Four principal types of proportion help solve problems involving ratios and their relationships.
## Core Concept
Proportion is a statement that two ratios are equal. When a:b = c:d, we say a, b, c, d are in proportion and write a:b::c:d. The four types classify how quantities relate to each other—critical for solving business problems involving scaling, pricing, resource allocation, and variation.
## The Four Types of Proportion
### 1. Simple (Direct) Proportion Two quantities are in direct proportion if one increases/decreases and the other increases/decreases in the same ratio. - Rule: If x ∝ y, then x = ky (where k = constant of proportionality) - Practical use: Cost and quantity; speed and distance (at constant time); wage and hours worked - Test: x₁/y₁ = x₂/y₂
### 2. Inverse (Indirect) Proportion Two quantities are in inverse proportion if one increases while the other decreases in the inverse ratio. - Rule: If x ∝ 1/y, then x = k/y or xy = k - Practical use: Time and workers; speed and time (fixed distance); price and demand (simplified model) - Test: x₁ × y₁ = x₂ × y₂
### 3. Compound Proportion A quantity is proportional to two or more other quantities simultaneously. - Rule: If z ∝ x and z ∝ y, then z ∝ xy, so z = kxy - Practical use: Work output (depends on workers AND hours); commission (depends on sales AND rate); interest (depends on principal AND time AND rate) - Combined inverse-direct: May involve both types (e.g., z ∝ x/y means z = k(x/y))
### 4. Continued Proportion Three or more quantities satisfy the ratio relation a:b = b:c. - Rule: If a, b, c are in continued proportion, then b² = ac (b is geometric mean) - Variation: Four quantities a, b, c, d in continued proportion when a:b = b:c = c:d - Practical use: Growth sequences; scaling dimensions
## Formula / Rule
| Type | Relationship | Equation | Test Condition | |------|--------------|----------|---| | Direct | Both increase together | x = ky | x₁y₂ = x₂y₁ | | Inverse | One increases, one decreases | xy = k | x₁y₁ = x₂y₂ | | Compound | Multiple factors | z = kx₁x₂.../y₁y₂... | Apply component rules | | Continued | Three in sequence | b² = ac | Middle term² = product of extremes |
## Common Exam Applications
- Production problem: If 12 workers complete 240 units in 5 days (direct + compound), find units by 18 workers in 8 days
- Pricing: If 5 kg of material costs ₹450, find cost of 13 kg (direct proportion)
- Time & Work: If 8 workers finish a job in 15 days, in how many days will 10 workers finish it? (inverse proportion)
- Continued proportion: Find b if 4, b, 9 are in continued proportion → b² = 4 × 9 = 36 → b = 6
## Worked Example
Problem: A factory produces 300 units with 10 machines working 8 hours/day. How many units with 15 machines working 10 hours/day?
Solution (compound proportion): - Output ∝ (machines × hours/day) - 300 = k(10 × 8) - k = 300/80 = 3.75 units per (machine·hour) - New output = 3.75 × (15 × 10) = 3.75 × 150 = 562.5 units
## Common Mistakes
- Confusing direct and inverse (test with actual numbers before deciding)
- Forgetting to apply proportionality to ALL factors in compound problems
- Assuming "both increase" always means direct (could be coincidental—verify ratio equality)
- In continued proportion, using the wrong term as middle value